§1. Basics of sets
Operations of sets
• Union A∪B := {x: x ∈ A or x ∈ B}
∪α∈ΛAα = {x : x ∈ Aα for some α ∈ Λ}
• Intersection A∩B := {x : x ∈ A and x ∈ B}
∩α∈ΛAα = {x : x ∈ Aα for all α ∈ Λ}
• Difference A\B := {x : x ∈ A and x 6∈ B}
• Complement Ac := X \A
•Symmetric difference A4B := (A\B)∪(B\A) Properties.
(1) A∪A = A, A∩A = A
(2) A4A= ∅, A4X = Ac, A4B = (A∪B)\(A∩B) (3) A∪ ∅ = A, A∩ ∅ = ∅, A\ ∅ = A, A4 ∅= A
(4) A∪B = B ∪A, A∩B = B ∩A, A4B = B 4A (5) A∪(B∪ C) = (A∪ B)∪C
A∩(B∩ C) = (A∩ B)∩C A4(B 4C) = (A4B)4C (6) A∩(B∪ C) = (A∩ B)∪(A∩C)
A∪(B∩ C) = (A∪ B)∩(A∪C) (7) A∩(B\C) = (A∩B)\(A∩ C)
(8) (A\B)\C = A\(B∪C), A\(B\C) = (A\B)∪(A∩C) (9) A∪B = (A∆B)∪ (A∩B) =A∪(B \A)
(10) A\(B ∪C) = (A\B)∩ (A\C) (11) A\(B ∩C) = (A\B)∪ (A\C)
Proposition. (de Morgan) Let S be a set and let{Aα : α ∈ Λ} be a family of sets. Then
S \ [
α∈Λ
Aα = \
α∈Λ
(S \Aα), S \ \
α∈Λ
Aα = [
α∈Λ
(S \Aα).
Rings and algebras
A ring, R, over a set X, is a class of subsets of X such that A∪ B, A \B ∈ R for all A, B ∈ R. An algebra F over X is a ring over X with X ∈ F; an algebra is also called a field.
Remarks. Let R be a ring over a set X.
(i) It is clear that ∅ ∈ R. In particular, R = {∅} when- ever X = ∅.
(ii) R is closed under the operation intersection.
Proposition. Let R be a class of subsets of a set X. Then R is a ring over X if and only if R is closed under operations of finitely many unions and differences.
Proposition. Let F be a class of subsets of a set X. The following statements are equivalent.
(i) F is an algebra over X.
(ii) F is closed under operations of union, difference and complement.
(iii) F is closed under operations of finitely many unions, differences and complements.
Examples.
(1) Let P(X) := {A : A ⊂ X}, then P(X) is the largest ring/algebra over X. {∅} and {∅, X} are the smallest ring and algebra over X, respectively.
(2) Let X = {1,2}. Then R1 =
∅,{1} and R2 =
∅,{2}
are both rings on X.
(3) Let F be the class of all finitely many unions of semi- closed “intervals” of the form (a, b], then F is a ring over R.
Note thatR1∪R2 is not a ring withR1,R2 given in above Example (2). However, R1 ∩ R2 is a ring. In general, we have the following
Proposition. Let M be a family of rings over X. Then R := \
R0∈M
R0
is a ring over X as well.
Proposition. Let M be a family of algebras over X. Then
F := \
F0∈M
F0
is an algebra over X as well.
Proposition. Let A ∈ P(X). Then there exists the smallest ring (or algebra) R (or F) over X such that A ∈ R (or A ⊂ F), that is to say, for every ring R0 (or every algebra F0) over X satisfying A ∈ R0 (or A ∈ F0) we have R ⊂ R0 (or F ⊂ F0).
Proof. Indeed, R = {∅, A} and F = {∅, A, Ac, X}.
Theorem. LetA ⊂ P(X). Then there exists the small- est ring (or algebra) R (or F) over X such that A ⊂ R (or A ⊂ F), that is to say, for every ring R0 (or every algebra F0) over X satisfying A ⊂ R0 (or A ⊂ F0) we have R ⊂ R0 (or F ⊂ F0).
Proof. Note that the set F defined by F := \
F0 : A ⊂ F0 ⊂ P(X) and F0 is an algebra
is an algebra over X. Clearly, F is the smallest algebra containing A as an element. Also,
R := \
R0 : A ⊂ R0 ⊂ P(X) and R0 is a ring
is the ring desired.
Remark. R (or F) given in above theorem is called the ring ( or algebra) generated by A, and we denote it by R(A) (or F(A)). In particular, R(∅) ={∅} and
R(X) = F(∅) ={∅, X}.
Limits of sets
Monotonic sequences of sets.
(1) {An} is increasing: An ⊂ An+1 for all n.
(2) {An} is decreasing: An ⊃An+1 for all n.
(3) {An} is monotone if it is either increasing or de- creasing.
Remark. Let {An} be decreasing. Then A1 = (A1\A2)∪(A2\A3)∪· · ·∪(An\An+1)∪· · ·∪
∞
\
n=1
An
; moreover, all terms are pairwise disjoint.
Limits of sets.
(1) limit superior:
n→∞lim An := lim sup
n→∞
An :=
∞
\
k=1
∞
[
n=k
An
(2) limit inferior:
lim
n→∞
An := lim inf
n→∞ An :=
∞
[
k=1
∞
\
n=k
An
(3) limit: We say that the sequence {An} of sets An is convergent if
lim sup
n→∞
An = lim inf
n→∞ An, and define the limit of {An} by
n→∞lim An := lim sup
n→∞
An = lim inf
n→∞ An.
Properties.
(1) lim
n→∞An = {x : x∈ An for infinitely many n}.
(2) lim
n→∞
An = {x : x∈ An for all but finitely many n}.
(3)
∞
\
n=1
An ⊂ lim
n→∞
An ⊂ lim
n→∞An ⊂
∞
[
n=1
An.
Proposition.
(1) If {An} is increasing, then
n→∞lim An =
∞
[
n=1
An.
(2) If {An} is decreasing, then
n→∞lim An =
∞
\
n=1
An.
Examples. Let n ∈ N+.
(1) Let An = [n,∞). Then limn→∞An = ∅.
(2) Let An = (−n1,n1). Then limn→∞An = {0}.
(3) Let An = (−1 + 1n,1− 1n), n ≥ 2. Then
n→∞lim An = (−1,1).
(4) Let A2n = [0,2− 2n+11 ], A2n+1 = [0,1 + 2n1 ]. Then
n→∞lim An = [0,2), lim
n→∞
An = [0,1].
(5) Let fn, f : R→ R. Then t: fn(t) → f(t) =
∞
\
k=1
∞
[
m=1
∞
\
n=m
t : |fn(t)−f(t)| < 1 k
, t: fn(t) 6→ f(t) =
∞
[
k=1
∞
\
m=1
∞
[
n=m
t : |fn(t)−f(t)| ≥ 1 k
.
Functions and sets.
Let f : E → R (or C). Write
E[f ≥ c] :={x ∈ E : f(x) ≥c}, E[f ≤ c] :={x ∈ E : f(x) ≤c}, E[f < c] :={x ∈ E : f(x) < c}, E[f > c] :={x ∈ E : f(x) > c}, E[f = c] :={x ∈ E : f(x) =c}.
It is clear that
(1) E[f ≥ c]∪ E[f < c] = E.
(2) E[f ≥ c]∩ E[f < c] = ∅.
(3) E[f > c]∩E[f ≤d] = E[c < f ≤ d].
(4) E[f ≤ c] = T∞
n=1E[f < c+ n1].
Proposition. Let fn → f as n → ∞ (i.e., pointwise).
Then
E[f ≤ c] =
∞
\
k=1
∞
[
N=1
∞
\
n≥N
E[fn ≤ c+ 1/k],
=
∞
\
k=1
lim
n→∞
E[fn ≤c+ 1/k].
If further f1 ≤ f2 ≤ · · · ≤fn ≤ fn+1 ≤ · · ·, then E[f ≤ c] =
∞
\
n=1
E[fn ≤c] = lim
n→∞E[fn ≤c].
Proof. It suffices to prove the first equality. Let x ∈
∞
\
k=1
∞
[
N=1
∞
\
n≥N
E[fn ≤ c+ 1/k].
Then, for every k ∈ N, there is an N ∈ N such that x ∈ E[fn ≤ c+ 1/k]
for all n ≥ N, i.e.,
fn(x) ≤ c+ 1/k, n ≥N.
By letting n → ∞ in both sides of above inequality, we obtain
f(x) ≤ c+ 1/k, k ∈ N.
Again, letting k → ∞ yields f(x) ≤ c. Thus, x ∈ E[f ≤c].
Conversely, let x ∈ E[f ≤c]. Then a := lim
n→∞fn(x) = f(x) ≤ c.
Thus, for every k ∈ N, there is an N ∈ N such that
|fn(x)−a| ≤ 1/k, n ≥ N, and hence,
fn(x)−c ≤ fn(x)−a ≤1/k, n≥ N.
This implies that x ∈
∞
\
k=1
∞
[
N=1
∞
\
n≥N
E[fn ≤ c+ 1/k].
Characteristic functions of sets.
Let X be a nonempty set and let A ⊂ X. We call the function χA (defined on X)
χA(x) :=
(1, x ∈ A;
0, x 6∈ A, the characteristic function of A.
We list some basic properties as follows, which can be proved directly by the definition of characteristic functions and the de Morgan law of sets.
(1) χA ≡ 1⇔ A = X; χA ≡0 ⇔A = ∅.
(2) χA ≤ χB ⇔A ⊂ B; χA = χB ⇔A = B. (3) Let U := S
α∈ΛAα and M := T
α∈ΛAα. Then χU(x) = max
α∈Λ χAα(x), χM(x) = min
α∈Λ χAα(x).
(4) Let {An} ⊂ X. Then, for every x ∈ X, χlim
n→∞An(x) = lim
n→∞χAn(x), χlimn→∞An(x) = lim
n→∞
χAn(x).
(5) Let {An} ⊂ X. Then {An} converges if and only if {χAn(x)} converges for every x ∈ X. Moreover,
χlimn→∞An(x) = lim
n→∞χAn(x), x ∈ X.
Proof. (2) Suppose that χA ≤ χB, i.e., χA(x) ≤ χB(x), x ∈ X.
Let x ∈ A. Then χB(x) ≥ χA(x) = 1, so that χB(x) = 1.
This implies that x ∈ B. Thus, A ⊂ B.
Conversely, suppose that A⊂ B.
(i) If x ∈ A, then x ∈ B, so that χA(x) = χB(x) = 1.
(ii) If x 6∈ A, then χA(x) = 0 ≤ χB(x).
In a word, we have χA ≤χB.
Analogously, (3) and (4) can be proved by the de Morgan law of sets. (5) is a direct consequence of (4).
§2. Mappings and Cardinalities of sets
Mappings.
Some notations.
Let A, B be nonempty sets. A mapping ϕ from A to B is a rule that assigns to each element x ∈ A a unique element ϕ(x) ∈ B. We call ϕ(x) the image of x under the mapping ϕ.
To indicate that ϕ is a mapping from A to B, we often write ϕ : A → B. The set A is called the domain of ϕ (denoted by D(ϕ)). The set ϕ(A) := {ϕ(x) : x ∈ A} is called the range of ϕ (also, denoted by R(ϕ)). Clearly, ϕ(A) ⊂ B.
For y ∈ B fixed, the set {x ∈ A : ϕ(x) = y} is called the inverse image of y under the mapping ϕ (denoted by ϕ−1(y)). In addition, for E ⊂ B fixed, the set {x ∈ A : ϕ(x) ∈ E} is called the inverse image of E under the mapping ϕ (denoted by ϕ−1(E)).
The mapping ϕ :A → B is said to be
(1) one-to-one (or injective) if ϕ(x1) = ϕ(x2) implies that x1 = x2.
(2) onto (or surjective) if ϕ(A) =B.
(3) 1-1 correspondence (or bijective) if ϕ is one-to- one and onto.
Remark. Clearly, an injective ϕ is bijective from D(ϕ) to R(ϕ).
Extensions of mappings. Let ϕ : D(ϕ) → B and ψ : D(ψ) → B. If D(ϕ) ⊂ D(ψ) and ψ(x) = ϕ(x) for all
x ∈ D(ϕ), then we call ψ is an extension of ϕ to D(ψ) (denoted by ϕ ⊂ ψ). If this is the case, we call ϕ the restriction of ψ on D(ϕ) (denoted by ϕ = ψ|D(ϕ)).
Compositions of mappings. Let ϕ1 : A → B and ϕ2 : B → C. The composition of ϕ2 with ϕ1, denoted by ϕ2 ◦ϕ1, is the mapping ϕ2 ◦ϕ1 : A→ C defined by
(ϕ2 ◦ϕ1)(x) =ϕ2(ϕ1(x)), x ∈ A.
Inverse of a mapping. Let ϕ : A → B be injective.
For y ∈ R(ϕ), let ϕ−1(y) be the unique x ∈ A such that ϕ(x) = y. The mapping ϕ−1 : R(ϕ) → A so defined is called the inverse of the mapping ϕ. In this sense, an injective is also said to be an invertible mapping.
Identity mapping. Let A be a nonempty set. The identity mappingonAis the mapping ϕ: A → Adefined by
ϕ(x) =x, x∈ A.
Equivalence of sets
Definition. Two non-empty sets are said to be equiva- lent if there is a 1-1 correspondence from one to the other.
We writeA ∼ B if setsAandB are equivalent. In addition, we define ∅ ∼ ∅.
Remark. Let A, B and C be three sets. It is clear that (i) A ∼ A (reflexive)
(ii) A ∼ B implies that B ∼ A (symmetric)
(iii) A ∼B and B ∼C implies that A ∼ C (transitive) Remark. Let ϕ : A →B be injective. Then A ∼ R(ϕ).
Examples. Let a, b ∈ R. Then
[0,1] ∼[a, b] ∼[a, b) ∼ (a, b] ∼ (a, b).
Theorem (F. Bernstein, 1898). Let A and B be two sets. IfAis equivalent to a subset ofB whileB is equivalent to a subset of A, then A and B are equivalent.
Outline of the proof. Suppose that A ∼ B1 ⊂ B and B ∼ A1 ⊂ A.
Step I. Since the relation ∼ has translation, it suffices to show A ∼A1.
Step II. Consider the following disjoint decompositions A = C1 ∪C2 ∪ · · · ∪Cn∪ · · · ,
A1 = C10 ∪C20 ∪ · · · ∪Cn0 ∪ · · · . If Cn ∼ Cn0 for every n ∈ N, then A ∼ A1.
Step III. Construct such decompositions.
Proof. Suppose that A ∼ B1 ⊂ B and B ∼ A1 ⊂ A via 1-1 correspondences ϕ1 and ϕ2, respectively. Write A0 :=
A, ϕ := ϕ2◦ϕ1 and denote An+2 := ϕ(An) for n∈ N. Then we obtain a decreasing series of sets
A ⊃A1 ⊃ A2 ⊃ · · · ⊃ An ⊃ · · · . Also,
A ∼ A2 ∼A4 ∼ · · · ; A1 ∼A3 ∼ A5 ∼ · · · due to the same mapping ϕ. Notice that
A = (A\A1)∪(A1 \A2)∪(A2 \A3)∪(A3 \A4)∪ · · · ∪D, A1 = (A1 \A2)∪(A2 \A3)∪(A3 \A4)∪(A4 \A5)∪ · · · ∪D,
where D = A1 ∩A2 ∩A3 ∩ · · ·; and notice that A\A1 ∼ A2 \A3
A2 \A3 ∼ A4 \A5
· · ·
A2n\A2n+1 ∼ A2n+2 \A2n+3
· · ·
by virtue of the same mappingϕ. This implies thatA∼ A1,
so that A∼ B due to A1 ∼B.
Corollary. Let C ⊂ B ⊂A. If C ∼ A then B ∼A.
Cardinalities of sets.
Definition. Let A and B are two sets. We define the expressions
A ≤ B (or B ≥ A) and A= B
to mean that A is equivalent to a subset of B and A is equivalent to B, respectively. In addition, we call A the cardinality of A.
Definition. We also write
A < B (or B > A) to mean that A≤ B but A 6= B.
Bernstein theorem. LetAandB be two sets. IfA ≤ B and B ≤ A then A= B.
Remark. By Bernstein theorem and Zermelo Ax- iom of Choice (introduced in the next section), for two sets A and B, exactly one of the following three statements holds:
A < B, A = B, A > B.
Examples.
1. {a1,· · · , an} ∼ {1,· · · , n}.
2. {0,2,4,· · · ,2n,· · · } ∼ {0,1,2,· · · , n,· · · }.
3. N∼ Z.
4. [0,1] ∼ [a, b], where a, b ∈ R and a < b.
Finite and infinite sets:
Definition. A setAis called afinite setifA ∼ {1,· · · , n}
for some n ∈ N. A set is called an infinite set if it is not finite.
Theorem. A set is infinite if and only if it is equivalent to some of its proper subsets, or equivalently, a set is finite if and only if it is not equivalent to any of its proper subsets.
Proof. Let A be an infinite set. We can take {a1, a2,· · · , an,· · · } ⊂ A.
Now consider the proper subset A\ {a1} and define a map- ping ϕ :A → A\ {a1} by
(i) ϕ(x) := x for x ∈ A\ {a1, a2,· · · , an,· · · }, (ii) ϕ(ak) := ak+1 for k = 1,2,· · ·.
Clearly, ϕ is a bijective. Thus, A ∼A\ {a1}.
Conversely, let B be a proper subset of A such that B ∼ A. It is clear that B 6= ∅ (WHY). Assume that A is finite.
Then, by definition of finite sets, there exists an N ∈ N such that {1,2,· · · , N} ∼ A. Thus, there is a bijective ϕ1 : {1,2,· · · , N} → A, and hence,
A = {ϕ1(1), ϕ1(2),· · · , ϕ1(N)}.
Note that
B = {ϕ(n1), ϕ(n2),· · · , ϕ(nk)}, 1 ≤ k < N.
Thus, any mapping from A to B is not an injective (other- wise, k = N). This contracts to that A ∼ B. Thus, A is
infinite.
Countable sets.
Definition. A set A is said to be countable if A ∼ N. Remark. For convenience, we write N = ℵ0. Thus, A = ℵ0 for every countable set A.
Properties.
1. Every infinite set has a countable subset.
2. Any subset of a countable set is finite or countable.
3. If A is countable and B is finite or countable, then A∪B is countable.
4. If An is countable for every n ∈ N, then S∞
n=1An is also countable.
5. If A is an infinite set and B is finite or countable, then A∪B ∼ A.
Proof. 1. Let A be an infinite set. Clearly, we can take a1 ∈ A. Since A\ {a1} is also infinite (WHY), we can take a2 ∈ A\ {a1}. Note that, if we take a1, a2,· · · , ak ∈ A, then we can also take ak+1 ∈ (A\ {a1, a2,· · · , ak}) due to the infinity of A. Thus, by induction, we obtain a countable subset {a1, a2,· · · , an,· · · } of A.
2. Let A be a countable set, and let B ⊂ A. It suffices to show that B is countable wheneverB is not finite. Suppose that B is not finite. Note that
ℵ0 ≤B ≤A = ℵ0,
where the first inequality follows from 1, immediately. This implies that B = ℵ0 by Bernstein theorem.
3. Let A be countable, and let B be finite or countable.
Write
A = {a1, a2,· · · , an,· · · }.
(i) If B is finite, then B = {b1, b2,· · · , bn}, so that the mapping ϕ defined by
ϕ(bk) := ak, k = 1,2,· · · , n;
ϕ(am) := am+n, m ∈ N,
is a bijective from A∪B to A. Thus, A ∼ B.
(ii) If B is countable, thenB = {b1, b2,· · · , bn,· · · }. Note that the mapping ϕ defined by
ϕ(an) := a2n,
ϕ(bn) := a2n−1, n ∈ N,
is a bijective from A∪ B to A. That is to say, A ∪ B is arrayed as
b1, a1, b2, a2,· · · , bn, an,· · · . Thus, A∪B is also countable.
4. Let An be countable for every n ∈ N. Suppose that An = {a1n, a2n,· · · , amn,· · · }. Then S∞
n=1An can be arrayed as
a11, a21, a31,· · · , am1 ,· · · a12, a22, a32,· · · , am2 ,· · · a13, a23, a33,· · · , am3 ,· · ·
· · · · a1n, a2n, a3n,· · · , amn,· · ·
· · · · Thus, S∞
n=1An is countable as well.
Finally, the statement 5 can be proved by arguments analogous to that given in the last theorem.
Examples of countable sets.
1. N, Z, Q.
2. {(x, y) ∈ R2 : x, y ∈ Q}.
Uncountable sets.
Definition. An infinite set A is called uncountable if A 6∼N.
Remark. Clearly, A > ℵ0 for every uncountable set A.
Example. (Cantor, 1874) [0,1] > ℵ0.
Proof. It is clear that [0,1] is infinite. Assume that [0,1] = ℵ0. Then we can array it as
[0,1] = {a1, a2,· · · , an,· · · }.
Trisect the interval [0,1] and take one, [b1, b01], of the three subintervals such that a1 6∈ [b1, b01]; Trisect the in- terval [b1, b01] and take one, [b2, b02], of the three subintervals such that a2 6∈ [b2, b02]; · · · ; Trisect the interval [bk, b0k] and take one, [bk+1, b0k+1], of the three subintervals such that ak+1 6∈ [bk+1, b0k+1]. Thus, we obtain a decreasing sequence {[bn, b0n]} with an 6∈ [bn, b0n] for every n ∈ N, and hence
{a1, a2,· · · , an,· · · } ∩ ∞
\
n=1
[bn, b0n]
6= ∅.
On the other hand, by the completeness of real numbers, there is a real number a ∈ [0,1] such that
∞
\
n=1
[bn, b0n] = {a}.
Thus, a 6= an for all n∈ N, so that
[0,1] 6= {a1, a2,· · · , an,· · · }.
This yields a contradiction. Thus, [0,1] > ℵ0.
Cardinality of the continuum. A set A is said to have the cardinality of the continuum if A ∼ [0,1]. And we denote it by ¯A¯= ℵ.
Example. Let a, b ∈ R and a < b. Then
[a, b] = [a, b) = (a, b] = (a, b) =R = ℵ.
Proposition. Let An = ℵ for every n ∈ N. Then
∞
[
n=1
An = ℵ.
Proof. It is clear that
∞
[
n=1
An ≥A1 = ℵ.
On the other hand, write
A01 := A1, A02 := A2 \A1,· · · , A0n := An \ [
k<n
Ak,· · · . Then {A0n} is a disjoint decomposition of S∞
n=1An. Note that we can map A01 to [1,2) via a bijective and A0k to [k, k+ 1) via an injective for every k ≥ 2. Thus, there is an injective from S∞
n=1A0n to [1,∞), and hence,
∞
[
n=1
An =
∞
[
n=1
A0n ≤[1,∞) =ℵ.
Examples.
1. R= ℵ.
2. R−Q= ℵ.
3. Rn = R∞ = ℵ.
4. Qn = ℵ0, Q∞ = ℵ.
5. {{an} : an = 0 or 1}= ℵ.
Theorem. (Cantor, 1891) Let A be a set. Then A < P(A).
Proof. Suppose to the contrary that there is a bijection φ from A to P(A). Note that φ(x) ∈ P(A). We can define
B = {x ∈ A: x 6∈ φ(x)},
then B ⊂ A, i.e., B ∈ P(A). Since φ is surjective, there is some x0 ∈ A such that φ(x0) = B. However, either x0 ∈ B
or x0 6∈ B will give contradiction.
Remark. Note that, if A = n, then P(A) = 2n. Thus, we write P(A) := 2A for short.
Problem. By Cantor theorem, N < 2N, i.e., ℵ0 < 2ℵ0. Also, we have proved that ℵ0 < ℵ. What about the cardi- nalities 2ℵ0 and ℵ?
Proposition. ℵ0 < P(N) = ℵ.
Proof. It suffices to show P(N) = ℵ. We decompose the proof into the following three steps.
(i) For the class
B := {B ⊂ N: B is finite}, we have B = ℵ0. Indeed, note that
B = [
n∈N
Bn,
where Bn denotes the class of subsets B with B = n. By the method of induction, it is easy to see that the class Bn is countable for every n ∈ N, so that B is countable.
(ii) For the class
C := {C ⊂ N: C is infinite},
we have C = ℵ. Indeed, for every x ∈ C and define the binary decimal ϕ(x) as
ϕ(x) := 0.a1a2· · ·an· · · ,
where an = 1 if n ∈ C, and an = 0 if n 6∈ C. Then ϕ is a bijective from C to the set of allbinary infinite decimals in (0,1], and hence, C = ℵ.
(iii) B ∪ C = ℵ. Indeed,
B ∪ C = (C \ B)∪ B.
Since B is countable, there is an injective from B to [0,1).
Also, sinceC \ B = ℵ (WHY), there is a bijective fromC \B to [1,2). Thus, there is an injective from (C \B)∪B to [0,2).
So we have
ℵ = C ≤ B ∪ C ≤ [0,2) = ℵ.
Remark. It is clear that
1< 2< · · · < n < n+ 1 < · · · < ℵ0 < 2ℵ0 < 22ℵ0 < · · · . Therefore, the set with maximal cardinality does not exist.
Example. R[a, b]> C[a, b] = ℵ.
Proof. Let{r1,· · · , rn,· · · } = [a, b]∩Q, then there is a one- to-one mapping between f ∈ C[a, b] to R∞ by
f 7→ {f(r1), f(r2),· · · , f(rn),· · · } ∈ R∞.
Therefore C[a, b] ≤ R∞ = ℵ. On the other hand, any of constant functions belongs to C[a, b]. Notice that the set, K, of all constant functions has cardinality ℵ, so that R∞ is equivalent to K, a subset of C[a, b]. This implies that R∞ ≤ C[a, b]. Therefore, C[a, b] = ℵ by Bernstein’s theorem.
On the other hand, for everyA ∈ P([0,1]), defineϕ(A) :=
χA. Then ϕ is an injective from P([0,1]) to R[a, b], so that 2ℵ = P([0,1])≤ R[a, b].
Also, for every f ∈ R[a, b], define
φ(f) := {(x, f(x)) : x ∈ [0,1]} ⊂ [0,1] ×R.
Then φ is an injective from R[a, b] to P([0,1]×R), so that R[a, b]≤ P([0,1]×R) = 2ℵ,
where the equality follows from the fact that [0,1]×R= ℵ
(WHY).
§3. Equivalence relations, orderings and axiom of choice
Equivalence relations. Let A be a nonempty set. A relation, ∼, on A is said to be an equivalence relation if for all x, y, z ∈ A,
• x ∼ x (reflexive)
• x ∼ y implies that y ∼ x (symmetric)
• x ∼ y and y ∼ z implies that x ∼z (transitive) Equivalence classes. Let A be a nonempty set and ∼ an equivalence relation on A. For each x ∈ A, define Ex :=
{y ∈ A : y ∼ x}. And let B := {Ex : x ∈ A}. Each member of B is called an equivalence class of A under
∼.
Properties. Let A be a nonempty set and ∼ an equiv- alence relation on A. Then
• for each x, y ∈ A, either Ex∩Ey = ∅ or Ex = Ey;
• A = S
x∈AEx;
• ∼ partitions A into disjoint equivalence classes, that is,Ais a disjoint union of the equivalence classes under ∼.
Ordering relations.
Definition. (Partial ordering) Let A be a set. A bi- nary relation defined between certain pairs (x, y) of ele- ments of A, expressed by x ≺ y, is called a partial order- ing on A if for all x, y, z ∈ A,
(i) (reflexivity) x ≺ x,
(ii) (antisymmetry) x ≺ y and y ≺ x implies that x = y,
(iii) (transitivity) x ≺y and y ≺ z implies that x ≺ z.
In addition, a set endowed with a partial ordering is called a partially ordered set.
Definition. (Total ordering) A partial ordering ≺ is called a total ordering if additionally,
(iv) for every pair (x, y) in A, either x ≺ y or y ≺x.
A set endowed with a total ordering is called a totally ordered set.
Examples.
(1) (P(A),⊂) is a partially ordered set.
(2) (R,≤) is a totally ordered set.
(3) Let Gbe a group. Let S be the set of subgroups with the relation that H ≺ H0 if H is a subgroup of H0. Then (S,≺) is a partially ordered set. Given two subgroups, H, H0 of G, we do not necessarily have H ≺H0 or H0 ≺ H.
Induced orderings. Let (A,≺) be a partially ordered set, and B a subset of A. We can define a partial ordering on B by defining thatx ≺ y for x, y ∈ B to hold if and only if x ≺ y in A. We shall say that it is the partial ordering on B induced by the ordering on A.
Upper bounds and maximal elements. Let (A,≺) be a partially ordered set, and B a subset of A. An upper boundof B (in A) is an elementx ∈ Asuch that y ≺ x for all y ∈ B. By a maximal element m of A one means an element of A such that if x ∈ A and m ≺ x, then m = x.
In addition, we call s ∈ A is a minimal element of A if s ≺ x for all comparable x ∈ A.
Zermelo’s axiom of choice.
Axiom of choice. Suppose that C is a collection of nonempty sets. Then there exists a mapping ϕ : C → S
A∈CA such that ϕ(A) ∈ A for each A ∈ C.
Remarks. (1) An equivalent statement of the Zermelo’s axiom of choice is: suppose thatC is a collection of nonempty sets, then there exists a set B such that B ⊂ S
A∈CA and A∩B has only one element for each A∈ C.
(2) Roughly speaking, the axiom of choice asserts that given a collection of nonempty sets, it is possible to select an element from each set in the collection.
(3) Although most mathematicians use the axiom of choice without hesitation, some employ it only when they cannot obtain a proof without it and others consider it unaccept- able. In real analysis and functional analysis, we will accept the axiom of choice and apply it freely.
Zorn’s lemma. (Principle of transfinite induction) Let A be a nonempty partially ordered set with the prop- erty that every totally ordered subset of A has an upper bound in X. Then A contains at least one maximal ele- ment.
Well ordering principle. Every set X has at least one well-ordering.
Trichotomy of cardinality. Let A and B be two sets.
Then exactly one of the following three statements holds:
A < B, A = B, A > B.
Continuum hypothesis. For every infinite cardinality m, there is no cardinality n such that m < n <2m.
Theorem. The following five results are equivalent.
(1) Axiom of choice.
(2) Zorn’s lemma.
(3) Well ordering principle.
(4) Trichotomy of cardinality.
(5) Continuum hypothesis.
Hilbert’s 23 problems - 1900.
1. Continuum hypothesis (G. Cantor, 1878):
• The contributions of K. G¨odel (1940) and P. Cohen (1963) showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo-Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent. However, there is no consensus on whether this is a solution to the problem.
• Recent work of W. H. Woodin (2010,2011) has raised
“hope” that there is an imminent solution.
• Is the Continuum Hypothesis a definite mathematical problem? – S. Feferman (2011)
2. The compatibility of the arithmetical axioms.
3. The equality of the volumes of two tetrahedra of equal bases and equal altitudes.
4. Problem of the straight line as the shortest distance between two points.
5. Lie’s concept of a continuous group of transformations without the assumption of the differentiability of the func- tions defining the group.
6. Mathematical treatment of the axioms of physics.
7. Irrationality and transcendence of certain numbers.
8. Problems of prime numbers.
9. Proof of the most general law of reciprocity in any num- ber field.
10. Determination of the solvability of a Diophantine equa- tion.
11. Quadratic forms with any algebraic numerical coeffi- cients.
12. Extension of Kroneckers theorem on abelian fields to any algebraic realm or rationality.
13. Impossibility of the solution of the general equation of the 7th degree by means of functions of only two arguments.
14. Proof of the finiteness of certain complete systems of functions.
15. Rigorous foundation of Schuberts enumerative calculus.
16. Problem of the topology of algebraic curves and sur- faces.
17. Expression of definite forms by squares.
18. Building up of space from congruent polyhedra.
19. Are the solutions of regular problems in the calculus of variations always necessarily analytic?
20. The general problem of boundary values.
21. Proof of the existence of linear differential equations having a prescribed monodromic group.
22. Uniformization of analytic relations by means of auto- morphic functions.
23. Further development of the methods of the calculus of variations.
§5. Point sets on the line
Intervals.
• (a, b) := {x : a < x < b}, −∞ ≤ a < b ≤ ∞
• [a, b) := {x : a ≤x < b}, −∞ < a < b≤ ∞
• (a, b] := {x : a < x ≤ b}, −∞ ≤ a < b < ∞
• [a, b] := {x: a ≤ x ≤ b}, −∞ < a ≤ b < ∞ In particular, [a, a] = {a}.
Bounded sets.
Definition. Let A be a nonempty subset of R. A real number c is called an upper bound (or lower bound) of A if x ≤ c (or c ≤ x) for all x ∈ A. If a subset of R has an upper bound (or lower bound), then we say that it is bounded above (or bounded below). A set is called bounded if it is bounded both above and below.
Definition. A real number u is called a least upper bound or supremum of A if it is an upper bound of A and smaller than or equal to any other upper bound of A.
We write u by
supA, sup
x∈A
x, or sup{x : x ∈ A}.
Definition. A real number l is called a greatest lower bound or infimum of A if it is a lower bound of A and greater than or equal to any other lower bound of A. We write l by
infA, inf
x∈Ax, or inf{x : x ∈ A}.
Remark. We define supA := +∞ if A is not bounded above, and define infA := −∞ if A is not bounded below;
In addition, (whenever needed) we may define sup∅ := −∞, inf∅:= +∞.
Open sets.
Definition. A subset O ⊂R is said to be anopen set if for eachx ∈ O, there is an r > 0 such that (x−r, x+r) ⊂O.
Remark. The open interval (x −r, x + r), denoted by N(x, r), is also called an r-neighborhood of x. Thus, A subset O ⊂ R is open if and only if for each point x ∈ O, there is an rx-neighborhood N(x, rx) ⊂ O.
Properties.
• ∅ and R are open sets.
•If A and B are open sets, then so isA∩B. (closed for finite intersection)
•If{Oα}α∈Λ is a collection of open sets, thenS
α∈ΛOα is open. (closed for arbitrary union)
Theorem. (constructions of open sets) Each open set O is countable union of disjoint open intervals. The representation is unique in the sense that if C and Dare two pairwise disjoint collections of open intervals whose union is O, then C = D.
Proof. Let O be an open set. For x ∈ O fixed, define Ax :={y :y < x and (y, x) ⊂O},
Bx :={z : z > x and (x, z) ⊂O}.
The sets Ax and Bx are nonempty because O is open. Let ax := inf Ax and bx := supBx. Then ax < x < bx and ax, bx 6∈ O.
Set Ix := (ax, bx) and note that x ∈ Ix. We claim that Ix ⊂O. Indeed, let u ∈ Ix; then ax < u < bx. Thus, we can choose y ∈ Ax and z ∈ Bx such that y < u < z. If u ≤ x, then u ∈ (y, x] ⊂ O and, if u > x, then u ∈ (x, z) ⊂ O.
Hence, Ix ⊂ O, so that S
x∈OIx ⊂ O. On the other hand, as x ∈ Ix, S
x∈OIx ⊃O. Thus, by write C := {Ix : x ∈ O}, we have O = S
A∈CA.